Ingeometry,Hermann–Mauguin notation is used to represent thesymmetry elements inpoint groups,plane groups andspace groups. It is named after the German crystallographerCarl Hermann (who introduced it in 1928) and the French mineralogistCharles-Victor Mauguin (who modified it in 1931). This notation is sometimes calledinternational notation, because it was adopted as standard by theInternational Tables For Crystallography since their first edition in 1935.

The Hermann–Mauguin notation, compared with theSchoenflies notation, is preferred incrystallography because it can easily be used to includetranslational symmetry elements, and it specifies the directions of the symmetry axes.^[1][2]

Rotation axes are denoted by a numbern – 1, 2, 3, 4, 5, 6, 7, 8, ... (angle of rotationφ =⁠360°/n⁠). Forimproper rotations, Hermann–Mauguin symbols show rotoinversion axes, unlikeSchoenflies andShubnikov notations, that shows rotation-reflection axes. The rotoinversion axes are represented by the corresponding number with amacron,n –1,2,3,4,5,6,7,8, ... .2 is equivalent to a mirror plane and usually notated as m. The direction of the mirror plane is defined as the direction perpendicular to it (the direction of the2 axis).

Hermann–Mauguin symbols show non-equivalent axes and planes in a symmetrical fashion. The direction of a symmetry element corresponds to its position in the Hermann–Mauguin symbol. If a rotation axisn and a mirror plane m have the same direction, then they are denoted as a fraction⁠n/m⁠ or n /m.

If two or more axes have the same direction, the axis with higher symmetry is shown. Higher symmetry means that the axis generates a pattern with more points. For example, rotation axes 3, 4, 5, 6, 7, 8 generate 3-, 4-, 5-, 6-, 7-, 8-point patterns, respectively.Improper rotation axes3,4,5,6,7,8 generate 6-, 4-, 10-, 6-, 14-, 8-point patterns, respectively. If a rotation and a rotoinversion axis generate the same number of points, the rotation axis should be chosen. For example, the⁠3/m⁠ combination is equivalent to6. Since6 generates 6 points, and 3 generates only 3,6 should be written instead of⁠3/m⁠ (not⁠6/m⁠, because6 already contains the mirror plane m). Analogously, in the case when both 3 and3 axes are present,3 should be written. However we write⁠4/m⁠, not⁠4/m⁠, because both 4 and4 generate four points. In the case of the⁠6/m⁠ combination, where 2, 3, 6,3, and6 axes are present, axes3,6, and 6 all generate 6-point patterns, as we can see on the figure in the right, but the latter should be used because it is a rotation axis – the symbol will be ⁠6/m⁠.

Finally, the Hermann–Mauguin symbol depends on the type^{[clarification needed]} of thegroup.

These groups may contain only two-fold axes, mirror planes, and/or an inversion center. These are thecrystallographic point groups 1 and1 (triclinic crystal system), 2, m, and⁠2/m⁠ (monoclinic), and 222,⁠2/m⁠⁠2/m⁠⁠2/m⁠, and mm2 (orthorhombic). (The short form of⁠2/m⁠⁠2/m⁠⁠2/m⁠ is mmm.) If the symbol contains three positions, then they denote symmetry elements in thex,y,z direction, respectively.

• First position –primary direction –z direction, assigned to the higher-order axis.
• Second position – symmetrically equivalentsecondary directions, which are perpendicular to thez-axis. These can be 2, m, or ⁠2/m⁠.
• Third position – symmetrically equivalenttertiary directions, passing betweensecondary directions^{[clarification needed]}. These can be 2, m, or ⁠2/m⁠.

These are the crystallographic groups 3, 32, 3m,3, and3⁠2/m⁠ (trigonal crystal system), 4, 422, 4mm,4,42m,⁠4/m⁠, and⁠4/m⁠⁠2/m⁠⁠2/m⁠ (tetragonal), and 6, 622, 6mm,6,6m2,⁠6/m⁠, and⁠6/m⁠⁠2/m⁠⁠2/m⁠ (hexagonal). Analogously, symbols of non-crystallographic groups (with axes of order 5, 7, 8, 9, ...) can be constructed. These groups can be arranged in the following table

It can be noticed that in groups with odd-order axesn andn the third position in symbol is always absent, because alln directions, perpendicular to higher-order axis, are symmetrically equivalent. For example, in the picture of a triangle all three mirror planes (S₀,S₁,S₂) are equivalent – all of them pass through one vertex and the center of the opposite side. For even-order axesn andn there are⁠n/2⁠ secondary directions and⁠n/2⁠ tertiary directions. For example, in the picture of a regular hexagon one can distinguish two sets of mirror planes – three planes go through two opposite vertexes, and three other planes go through the centers of opposite sides. In this case any of two sets can be chosen assecondary directions, the rest set will betertiary directions. Hence groups42m,62m,82m, ... can be written as4m2,6m2,8m2, ... . For symbols of point groups this order usually doesn't matter; however, it will be important for Hermann–Mauguin symbols of corresponding space groups, where secondary directions are directions of symmetry elements along unit cell translationsb andc, while the tertiary directions correspond to the direction between unit cell translationsb andc. For example, symbols P6m2 and P62m denote two different space groups. This also applies to symbols of space groups with odd-order axes 3 and3. The perpendicular symmetry elements can go along unit cell translationsb andc or between them. Space groups P321 and P312 are examples of the former and the latter cases, respectively.

The symbol of point group3⁠2/m⁠ may be confusing; the correspondingSchoenflies symbol isD_3d, which means that the group consists of 3-fold axis, three perpendicular 2-fold axes, and 3 vertical diagonal planes passing between these 2-fold axes, so it seems that the group can be denoted as 32m or 3m2. However, one should remember that, unlike Schoenflies notation, the direction of a plane in a Hermann–Mauguin symbol is defined as the direction perpendicular to the plane, and in theD_3d group all mirror planes are perpendicular to 2-fold axes, so they should be written in the same position as⁠2/m⁠. Second, these⁠2/m⁠ complexes generate an inversion center, which combining with the 3-fold rotation axis generates a3 rotoinversion axis.

Groups withn = ∞ are called limit groups orCurie groups.

These are the crystallographic groups of acubic crystal system: 23, 432,⁠2/m⁠3,43m, and⁠4/m⁠3⁠2/m⁠. All of them contain four diagonal 3-fold axes. These axes are arranged as 3-fold axes in a cube, directed along its four space diagonals (the cube has⁠4/m⁠3⁠2/m⁠ symmetry). These symbols are constructed the following way:

• First position – symmetrically equivalent directions of the coordinate axesx,y, andz. They are equivalent due to the presence of diagonal 3-fold axes.
• Second position – diagonal 3 or3 axes.
• Third position – diagonal directions between any two of the three coordinate axesx,y, andz. These can be 2, m, or⁠2/m⁠.

All Hermann–Mauguin symbols presented above are calledfull symbols. For many groups they can be simplified by omittingn-fold rotation axes in⁠n/m⁠ positions. This can be done if the rotation axis can be unambiguously obtained from the combination of symmetry elements presented in the symbol. For example, theshort symbol for⁠2/m⁠⁠2/m⁠⁠2/m⁠ is mmm, for⁠4/m⁠⁠2/m⁠⁠2/m⁠ is⁠4/m⁠mm, and for⁠4/m⁠3⁠2/m⁠ is m3m. In groups containing one higher-order axis, this higher-order axis cannot be omitted. For example, symbols⁠4/m⁠⁠2/m⁠⁠2/m⁠ and⁠6/m⁠⁠2/m⁠⁠2/m⁠ can be simplified to 4/mmm (or⁠4/m⁠mm) and 6/mmm (or⁠6/m⁠mm), but not to mmm; the short symbol for3⁠2/m⁠ is3m. The full and short symbols for all 32 crystallographic point groups are given incrystallographic point groups page.

Besides five cubic groups, there are two more non-crystallographic icosahedral groups (I andI_h inSchoenflies notation) and two limit groups (K andK_h inSchoenflies notation). The Hermann–Mauguin symbols were not designed for non-crystallographic groups, so their symbols are rather nominal and based on similarity to symbols of the crystallographic groups of a cubic crystal system.^{[3][4][5][6][7]} GroupI can be denoted as 235, 25, 532, 53. The possible short symbols forI_h are m35, m5, m5m,53m. The possible symbols for limit groupK are ∞∞ or 2∞, and forK_h are⁠∞/m⁠∞ or m∞ or ∞∞m.

Plane groups can be depicted using the Hermann–Mauguin system. The first letter is either lowercasep orc to represent primitive or centeredunit cells. The next number is the rotational symmetry, as given above. The presence of mirror planes are denotedm, whileglide reflections are only denotedg.Screw axes do not exist in two-dimensional spaces.

The symbol of aspace group is defined by combining the uppercase letter describing thelattice type with symbols specifying the symmetry elements. The symmetry elements are ordered the same way as in the symbol of corresponding point group (the group that is obtained if one removes all translational components from the space group). The symbols for symmetry elements are more diverse, because in addition to rotations axes and mirror planes, space group may contain more complex symmetry elements – screw axes (combination of rotation and translation) and glide planes (combination of mirror reflection and translation). As a result, many different space groups can correspond to the same point group. For example, choosing different lattice types and glide planes one can generate 28 different space groups from point group mmm, e.g. Pmmm, Pnnn, Pccm, Pban, Cmcm, Ibam, Fmmm, Fddd, and so on. In some cases, a space group is generated when translations are simply added to a point group.^[8] In other cases there is no point around which the point group applies. The notation is somewhat ambiguous, without a table giving more information. For example, space groups I23 and I2₁3 (nos. 197 and 199) both contain two-fold rotational axes as well as two-fold screw axes. In the first, the two-fold axes intersect the three-fold axes, whereas in the second they do not.^[9]

These are theBravais lattice types in three dimensions:

• P – Primitive
• I – Body centered (from the German "Innenzentriert")
• F – Face centered (from the German "Flächenzentriert")
• A – Base centered on A faces only
• B – Base centered on B faces only
• C – Base centered on C faces only
• R – Rhombohedral

Primitive,P	Base centered,C	Face centered,F	Body centered,I	Rhombohedral in hexagonal setting,R

Thescrew axis is noted by a number,n, where the angle of rotation is⁠360°/n⁠. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2₁ is a 180° (twofold) rotation followed by a translation of⁠1/2⁠ of the lattice vector. 3₁ is a 120° (threefold) rotation followed by a translation of⁠1/3⁠ of the lattice vector.

The possible screw axes are: 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, and 6₅.There are 4enantiomorphic pairs of axes: (3₁ – 3₂), (4₁ – 4₃), (6₁ – 6₅), and (6₂ – 6₄). This enantiomorphism results in 11 pairs of enantiomorphic space groups, namely

The orientation of aglide plane is given by the position of the symbol in the Hermann–Mauguin designation, just as with mirror planes.They are noted bya,b, orc depending on which axis (direction) the glide is along. There is also then glide, which is a glide along the half of a diagonal of a face, and thed glide, which is along a quarter of either a face or space diagonal of the unit cell. Thed glide is often called the diamond glide plane as it features in thediamond structure. In cases where there are two possibilities amonga,b, andc (such asa orb), the lettere is used. (In these cases, centering entails that both glides occur.) To summarize:

• a,b, orc glide translation along half the lattice vector of this face.
• n glide translation along half a face diagonal.
• d glide planes with translation along a quarter of a face diagonal or of a space diagonal.
• e two glides with the same glide plane and translation along two (different) half-lattice vectors.

Within the Hermann–Mauguin notation, operations involvingtime reversal—used indichromatic symmetry ormagnetic space group theory—are denoted by theprime symbol (′). For example, theidentity operation combined with time reversal is written as1′, amirror reflection combined with time reversal is denoted bym′, and ann-foldrotation axis combined with time reversal is expressed asn′ (e.g.,2′,4′, etc.).^[10]

• Decoding the Hermann-Maguin Notation – An introduction into the Hermann-Maguin notation for beginners.

• Crystallography
• Chemical nomenclature

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles needing clarification from January 2019
• Wikipedia articles needing clarification from August 2018